Question

For a given population with o = 10.5 lb. we want to test the null hypothesis j = 66.5 against the alternative hypothesis u # 66.5 on the basis of a random sample of size n = 64. If the null hypothesis is rejected when x < 64.6 lb. or å > 68.8. a) (3 points) What is the probability of a type l error? b) (4 points) What is the probability of a type II error and the power of the test when in reality u = 67.0?

Answer 1

a)

P(Typel Error) = P[Reject Ho|Hotrue] = 1 – P[Accept Ho|Hotrue] 64.6 – 66.5 68.8 - 66.5 =1 - P[64.6 < X < 68.8u = 66.5] = 1 - PL <25 V64 V64 = 1- P[-1.45 << < 1.75] = 1 – [P(Z < 1.75) - P(Z < -1.45)] 10.5 10.5 = 1-[P(Z < 1.75) - P(Z > 1.45)] = 1- [P(Z < 1.75) - [1 – P(Z < 1.45)]] = 1 – [0.95994 – [1 – 0.92647]] = 0.11359

b)

P(TypeII Error) = P[.Accept Ho|Hitrue) 10.5 10.5 64.6 - 67 68.8 – 67 = P[64.6 < X < 68.8|u = 67] = PL <<< 64 V64 = P[-1.83 <3 < 1.37] = P(Z < 1.37) - P(Z < -1.83) = P(Z < 1.37) - P(Z > 1.83) = P(Z < 1.37) - [1 – P(Z <1.83)] 0.91466 – [1 – 0.96638] = 0.88104